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cx^2-6cx+3x=15-5c la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2                         
c*x  - 6*c*x + 3*x = 15 - 5*c
3x+(cx2−6cx)=15−5c3 x + \left(c x^{2} - 6 c x\right) = 15 - 5 c
Simplificar
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.

La ecuación se convierte de
3x+(cx2−6cx)=15−5c3 x + \left(c x^{2} - 6 c x\right) = 15 - 5 c
en
(5c−15)+(3x+(cx2−6cx))=0\left(5 c - 15\right) + \left(3 x + \left(c x^{2} - 6 c x\right)\right) = 0
Es la ecuación de la forma
a*x^2 + b*x + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=ca = c
b=3−6cb = 3 - 6 c
c=5c−15c = 5 c - 15
, entonces
D = b^2 - 4 * a * c = 

(3 - 6*c)^2 - 4 * (c) * (-15 + 5*c) = (3 - 6*c)^2 - 4*c*(-15 + 5*c)

La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

o
x1=6c+−4c(5c−15)+(3−6c)2−32cx_{1} = \frac{6 c + \sqrt{- 4 c \left(5 c - 15\right) + \left(3 - 6 c\right)^{2}} - 3}{2 c}
x2=6c−−4c(5c−15)+(3−6c)2−32cx_{2} = \frac{6 c - \sqrt{- 4 c \left(5 c - 15\right) + \left(3 - 6 c\right)^{2}} - 3}{2 c}
Resolución de la ecuación paramétrica
Se da la ecuación con parámetro:
cx2−6cx+3x=15−5cc x^{2} - 6 c x + 3 x = 15 - 5 c
Коэффициент при x равен
cc
entonces son posibles los casos para c :
c<0c < 0
c=0c = 0
Consideremos todos los casos con detalles:
Con
c<0c < 0
la ecuación será
−x2+9x−20=0- x^{2} + 9 x - 20 = 0
su solución
x=4x = 4
x=5x = 5
Con
c=0c = 0
la ecuación será
3x−15=03 x - 15 = 0
su solución
x=5x = 5
Teorema de Cardano-Vieta
reescribamos la ecuación
3x+(cx2−6cx)=15−5c3 x + \left(c x^{2} - 6 c x\right) = 15 - 5 c
de
ax2+bx+c=0a x^{2} + b x + c = 0
como ecuación cuadrática reducida
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
cx2−6cx+5c+3x−15c=0\frac{c x^{2} - 6 c x + 5 c + 3 x - 15}{c} = 0
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=3−6ccp = \frac{3 - 6 c}{c}
q=caq = \frac{c}{a}
q=5c−15cq = \frac{5 c - 15}{c}
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−3−6ccx_{1} + x_{2} = - \frac{3 - 6 c}{c}
x1x2=5c−15cx_{1} x_{2} = \frac{5 c - 15}{c}
Gráfica
Suma y producto de raíces [src] 
suma
                                                      2                            
      /  im(c)*re(c)     (-3 + re(c))*im(c)\        im (c)       (-3 + re(c))*re(c)
5 + I*|--------------- - ------------------| + --------------- + ------------------
      |  2        2         2        2     |     2        2         2        2     
      \im (c) + re (c)    im (c) + re (c)  /   im (c) + re (c)    im (c) + re (c)
(i(−(re⁡(c)−3)im⁡(c)(re⁡(c))2+(im⁡(c))2+re⁡(c)im⁡(c)(re⁡(c))2+(im⁡(c))2)+(re⁡(c)−3)re⁡(c)(re⁡(c))2+(im⁡(c))2+(im⁡(c))2(re⁡(c))2+(im⁡(c))2)+5\left(i \left(- \frac{\left(\operatorname{re}{\left(c\right)} - 3\right) \operatorname{im}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} + \frac{\operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(c\right)} - 3\right) \operatorname{re}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(c\right)}\right)^{2}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}}\right) + 5 
=
                                                      2                            
      /  im(c)*re(c)     (-3 + re(c))*im(c)\        im (c)       (-3 + re(c))*re(c)
5 + I*|--------------- - ------------------| + --------------- + ------------------
      |  2        2         2        2     |     2        2         2        2     
      \im (c) + re (c)    im (c) + re (c)  /   im (c) + re (c)    im (c) + re (c)
i(−(re⁡(c)−3)im⁡(c)(re⁡(c))2+(im⁡(c))2+re⁡(c)im⁡(c)(re⁡(c))2+(im⁡(c))2)+(re⁡(c)−3)re⁡(c)(re⁡(c))2+(im⁡(c))2+5+(im⁡(c))2(re⁡(c))2+(im⁡(c))2i \left(- \frac{\left(\operatorname{re}{\left(c\right)} - 3\right) \operatorname{im}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} + \frac{\operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(c\right)} - 3\right) \operatorname{re}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} + 5 + \frac{\left(\operatorname{im}{\left(c\right)}\right)^{2}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} 
producto
  /                                                  2                            \
  |  /  im(c)*re(c)     (-3 + re(c))*im(c)\        im (c)       (-3 + re(c))*re(c)|
5*|I*|--------------- - ------------------| + --------------- + ------------------|
  |  |  2        2         2        2     |     2        2         2        2     |
  \  \im (c) + re (c)    im (c) + re (c)  /   im (c) + re (c)    im (c) + re (c)  /
5(i(−(re⁡(c)−3)im⁡(c)(re⁡(c))2+(im⁡(c))2+re⁡(c)im⁡(c)(re⁡(c))2+(im⁡(c))2)+(re⁡(c)−3)re⁡(c)(re⁡(c))2+(im⁡(c))2+(im⁡(c))2(re⁡(c))2+(im⁡(c))2)5 \left(i \left(- \frac{\left(\operatorname{re}{\left(c\right)} - 3\right) \operatorname{im}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} + \frac{\operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(c\right)} - 3\right) \operatorname{re}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(c\right)}\right)^{2}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}}\right) 
=
  /  2                                    \
5*\im (c) + (-3 + re(c))*re(c) + 3*I*im(c)/
-------------------------------------------
                2        2                 
              im (c) + re (c)
5((re⁡(c)−3)re⁡(c)+(im⁡(c))2+3iim⁡(c))(re⁡(c))2+(im⁡(c))2\frac{5 \left(\left(\operatorname{re}{\left(c\right)} - 3\right) \operatorname{re}{\left(c\right)} + \left(\operatorname{im}{\left(c\right)}\right)^{2} + 3 i \operatorname{im}{\left(c\right)}\right)}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} 
5*(im(c)^2 + (-3 + re(c))*re(c) + 3*i*im(c))/(im(c)^2 + re(c)^2)
Respuesta rápida [src] 
x1 = 5
x1=5x_{1} = 5 
                                                       2                            
       /  im(c)*re(c)     (-3 + re(c))*im(c)\        im (c)       (-3 + re(c))*re(c)
x2 = I*|--------------- - ------------------| + --------------- + ------------------
       |  2        2         2        2     |     2        2         2        2     
       \im (c) + re (c)    im (c) + re (c)  /   im (c) + re (c)    im (c) + re (c)
x2=i(−(re⁡(c)−3)im⁡(c)(re⁡(c))2+(im⁡(c))2+re⁡(c)im⁡(c)(re⁡(c))2+(im⁡(c))2)+(re⁡(c)−3)re⁡(c)(re⁡(c))2+(im⁡(c))2+(im⁡(c))2(re⁡(c))2+(im⁡(c))2x_{2} = i \left(- \frac{\left(\operatorname{re}{\left(c\right)} - 3\right) \operatorname{im}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} + \frac{\operatorname{re}{\left(c\right)} \operatorname{im}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(c\right)} - 3\right) \operatorname{re}{\left(c\right)}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(c\right)}\right)^{2}}{\left(\operatorname{re}{\left(c\right)}\right)^{2} + \left(\operatorname{im}{\left(c\right)}\right)^{2}} 
x2 = i*(-(re(c) - 3)*im(c)/(re(c)^2 + im(c)^2) + re(c)*im(c)/(re(c)^2 + im(c)^2)) + (re(c) - 3)*re(c)/(re(c)^2 + im(c)^2) + im(c)^2/(re(c)^2 + im(c)^2)
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